1. Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy Gustavo Sánchez April 2009

2. Summary VEC and Cointegrating VAR Models VEC and Cointegrating VAR Models Estimate Parameters Estimate Parameters Probability Forecasting Probability Forecasting Simulate Forecasts Simulate Forecasts Summary Statistics to estimate Summary Statistics to estimate probabilities of events

3. Point Forecast and Confidence Interval

4. Probability of Inflation Greater than 45 Proportion estimationNumber of obs = 225 ProportionStd. Err.[95% Conf. Interval] inf_45 00.28888890.03028380.22921130.348566 10.71111110.03028380.65143360.770789

5. Cointegrating VAR models Based on the vector error correction (VEC) model specification. Based on the vector error correction (VEC) model specification. The specification assumes that the economic theory characterizes the long-run equilibrium behavior The specification assumes that the economic theory characterizes the long-run equilibrium behavior The short-run fluctuations represent deviations from that equilibrium. The short-run fluctuations represent deviations from that equilibrium. The short-run and long-run (economic) concepts are linked to the statistical concept of stationarity. The short-run and long-run (economic) concepts are linked to the statistical concept of stationarity.

6. Cointegrating VAR models Reduced form for a VEC model I(1) Endogenous variables Where: Matrix with coefficients associated to short-run dynamic effects Vectors with coeficients associated to the intercepts and trends Vector with innovations Matrices containing the long-run adjustment coefficients and coefficients for the cointegrating relationships

7. Cointegrating VAR models Reduced form for a VEC model Identifying α and β requires r 2 restrictions Identifying α and β requires r 2 restrictions (r: number of cointegrating vectors). (r: number of cointegrating vectors). Johansen FIML estimation identifies α and β by imposing r 2 atheoretical restrictions. Johansen FIML estimation identifies α and β by imposing r 2 atheoretical restrictions.

8. Cointegrating VAR models Garrat et al. (2006) describe the Cointegrating VAR approach: Garrat et al. (2006) describe the Cointegrating VAR approach: Use economic theory to impose restrictions to identify αβ. Use economic theory to impose restrictions to identify αβ. Exact identification is not necessarily achieved by the theoretical restrictions. Exact identification is not necessarily achieved by the theoretical restrictions. Test whether the overidentifying restrictions are valid. Test whether the overidentifying restrictions are valid.

9. ** Restrictions on VEC system ** *** Restrictions on Beta lm1 *** constraint 1 [_ce1]lm1=1...... constraint 6 [_ce1]ltipp906bn=0 *** Restrictions on Beta lmt *** constraint 8 [_ce2]lmt=1...... constraint 11 [_ce2]ltipp906bn=0 *** Restrictions on alpha *** constraint 12 [D_loilp]l._ce1=0 constraint 13 [D_loilp]l._ce2=0 ** VEC specification ** vec lm1 lmt lcpi loilp ltcpn lxt ltipp906bn lgdp/// if tin(1991q1,2008Q4), lags(2) rank(2)/// bconstraints(1/11) aconstraints(12/13)/// noetable

10. Vector error-correction model Sample: 1991q1 - 2008q4 No. of obs = 72 AIC = -15.80442 Log likelihood = 659.9591 HQIC = -14.6589 Det(Sigma_ml) = 1.51e-18 SBIC = -12.92697 Cointegrating equations Equation Parms chi2 P>chi2 ------------------------------------------- _ce1 2 50.19532 0.0000 _ce2 3 1639.412 0.0000 ------------------------------------------- Identification: beta is overidentified Identifying constraints: ( 1) [_ce1]lm1 = 1 ( 2) [_ce1]lmt = 0 ( 3) [_ce1]lxt = 0 ( 4) [_ce1]loilp = 0 ( 5) [_ce1]lcpi = 0 ( 6) [_ce1]ltipp906bn = 0 ( 7) [_ce2]lm1 = 0 ( 8) [_ce2]lmt = 1 ( 9) [_ce2]lxt = 0 (10) [_ce2]ltcpn = 0 (11) [_ce2]ltipp906bn = 0

11. ------------------------------------------------------------------------------ beta | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _ce1 | lm1 | 1..... lmt | (dropped) lcpi | (dropped) loilp | (dropped) ltcpn |.215578.0697673 3.09 0.002.0788365.3523194 lxt | (dropped) ltipp906bn | (dropped) lgdp | -4.554976.6489147 -7.02 0.000 -5.826825 -3.283127 _cons | 57.02687..... -------------+---------------------------------------------------------------- _ce2 | lm1 | (dropped) lmt | 1..... lcpi | -.0317544.0087879 -3.61 0.000 -.0489784 -.0145304 loilp | -.0780758.0255611 -3.05 0.002 -.1281746 -.027977 ltcpn | (dropped) lxt | (dropped) ltipp906bn | (dropped) lgdp | -2.519458.1105036 -22.80 0.000 -2.736041 -2.302875 _cons | 26.26122..... ------------------------------------------------------------------------------

12. *** Point Forecast *** fcast compute y_, step(4) keepy_lm1 y_lmt y_lcpi/// y_loilp y_ltcpn y_lxt/// y_ltipp906bn y_lgdp quarter keep if tin(2009q1,2009q4) save "filename"

13. ** Residuals from the VEC equations ** foreach x of varlist lm1 lmt lxt loilp/// ltcpn lcpi/// ltipp906bn lgdp{ predict res_`x' if e(sample),/// residuals/// equation(D_`x') }

14. Probability Forecasting It is basically an estimation of the probability that a single or joint event occurs. It is basically an estimation of the probability that a single or joint event occurs. We could define the event in terms of the levels of one or more variables, for one or more future time periods. We could define the event in terms of the levels of one or more variables, for one or more future time periods. It is associated to the uncertainty inherent to the predictions produced by regression models. It is associated to the uncertainty inherent to the predictions produced by regression models.

15. Probability Forecasting This methodology can be applied to a wide diversity of models. Our focus here is on the predictions from a cointegrating VAR model. This methodology can be applied to a wide diversity of models. Our focus here is on the predictions from a cointegrating VAR model. In general, forecasting based on econometric models are subject to: In general, forecasting based on econometric models are subject to: Future uncertainty Future uncertainty Parameters uncertainty Parameters uncertainty Model uncertainty Model uncertainty Measurement and policy uncertainty Measurement and policy uncertainty

16. Probability Forecasting Future and parameter uncertainty Future and parameter uncertainty Let’s consider the standard linear regression model: Let’s consider the standard linear regression model: Where

17. Probability Forecasting Future and parameter uncertainty Future and parameter uncertainty For example, for σ 2 known we could simulate For example, for σ 2 known we could simulate ; j=1,2,…,J ; s=1,2,…,S Where: j-th random draw from s-th random draw from which is independent from the random draw for

18. Probability Forecasting Computations for VAR cointegrating models Computations for VAR cointegrating models Let’s consider the VEC model Let’s consider the VEC model Non-Parametric Approach 1.Simulated errors are drawn from in sample residuals 2. The Choleski decomposition for the estimated Var-Cov matrix of the error term is used in a two-stage procedure combined with the simulated errors in (1).

19. ** Matrix for Simulation (First Stage, Pag.167 ) ** matrix sigma=e(omega) /* V-C Matrix of the residuals */ matrix P=cholesky(sigma) mkmat res_lm1 res_lmt res_lxt res_loilp /// res_ltcpn res_lcpi /// res_lgdp res_ltipp906bn /// if tin(1991q1,2008q4),/// matrix(res) matrix invP_res=inv(P)*res' matrix invP_rs1=invP_res‘ svmat invP_rs1,names(col)

20. ** Program for Residual Resampling ** program mysim_np, rclass preserve bsample 4 if tin(1991q1,2008q4) /* 4 frcst. per. */ mkmat IP_R_D_lm1 IP_R_D_lm IP_R_D_lcpi/// IP_R_D_loilp IP_R_D_ltcpn IP_R_D_lxt/// IP_R_D_ltipp906bn IP_R_D_lgdp,/// matrix(IP_R) matrix PE_tr=P*IP_R' matrix PE=PE_tr' svmat PE,names(col) ● ● ●

21. ****** Simulation ****** simulate “varlist", rep(###)/// saving("filename",replace):/// mysim_np command: mysim_np s_lm1_1: r(res_lm1_1) s_lm1_2: r(res_lm1_2) ● ● ● ● ● ● ● ● ● s_lgdp_3: r(res_lgdp_3) s_lgdp_4: r(res_lgdp_4) Simulations (###) ─┼─ 1 ─┼─ 2 ─┼─ 3 ─┼─ 4 ─┼─ 5.................................................... 50 ● ● ● ● ● ● ● ● ●

22. **** Probability Forecasting **** generate dgdp=gdp/gdp2008*100-100/// if year==2009 & /// replication>0 generate inf=cpi/cpi2008*100-100 /// if year==2009 & /// replication>0 generate gdp_n__inf45=cond(dgdp 45,1,0) proportion gdp_n__inf35

23. Probability of Negative GDP and Inflation>45 Proportion estimationNumber of obs = 225 ProportionStd. Err.[95% Conf. Interval] gdp_1__inf45 0.68.0311677.6185805.7414195 1.32.0311677.2585805.3814195

24. Cointegrating VAR Models and Probability Forecasting: Applied to a Small Open Economy Gustavo Sánchez April 2009